A man had in his
office three cupboards, each
containing nine lockers, as shown in the diagram.
He told his clerk to
place a different one-figure number on each locker of cupboard A, and
to do the same in the case of B, and of C.
As we are here allowed to
call nought a digit, and he was not prohibited from using nought as a
number, he clearly had the option of omitting any one of ten digits
from each cupboard.

Now, the employer
did not say the lockers were to
be numbered in any numerical order, and he was surprised to find, when
the work was done, that the figures had apparently been mixed up
indiscriminately.
Calling upon his clerk for an explanation, the
eccentric lad stated that the notion had occurred to him so to arrange
the figures that in each case they formed a simple addition sum, the
two upper rows of figures producing the sum in the lowest row.
But the
most surprising point was this: that he had so arranged them that the
addition in A gave the smallest possible sum, that the addition in C
gave the largest possible sum, and that all the nine digits in the
three totals were different.

The puzzle is to
show how this could be
done.
No decimals are allowed and the nought may not appear in the
hundreds place.